The Plane Strain Young’s Modulus in Cubic Materials

Abstract

The orientation dependence of the plane strain Young’s modulus, E˜ , of cubic materials has been analysed as a function of the direction along which a uniaxial stress is applied to a single crystal and the perpendicular direction in the single crystal along which the strain is constrained to be zero. The locus of E˜ in the plane perpendicular to the axis of uniaxial stress is shown to be a circle when this stress is applied along 〈 111 〉. For materials with anisotropy ratios A> 1 , global minima in E˜ occur when the stress is applied along 〈 001 〉 and when the strain along one of the two perpendicular 〈 100 〉 directions is set to zero. Identical global maxima in E˜ are found when the stress is applied along two different families of 〈 uuw〉 directions and the direction of zero strain is along either a perpendicular 〈 1 1 ¯ 0 〉 or 〈 ww2 u‾ 〉 direction. For materials with A< 1 , the global maxima in E˜ occur when the stress is applied along 〈 001 〉 and when the strain along one of the two perpendicular 〈 100 〉 directions is set to zero, and identical global minima are found when the stress is applied along two different families of 〈 uuw〉 directions and the direction of zero strain is along either a perpendicular 〈 1 1 ¯ 0 〉 or 〈 ww2 u‾ 〉 direction.